Method for Determining Amplitude and Phase of Stratified Current of Overhead Wire

ABSTRACT

The present invention discloses a method for determining the amplitude and phase of a stratified current of an overhead wire, the method comprising the following steps: S 1 , determining the specification, the size and main technical parameters of a wire; S 2 , calculating mutual inductances between conductors within a single-phase wire and the self-inductance thereof; S 3 , calculating mutual inductance reactance between conductors within the single-phase wire of a three-phase system and the self-inductance reactance thereof; and S 4 , calculating the distribution of currents in each layer. The method takes into account the magnetic field coupling effect between conductors within a wire, so as to accurately calculate the current flowing through conductors in each layer within the wire, and accurately reflect a phase relationship between conductors in each layer.

TECHNICAL FIELD

The present invention relates to the technical field of calculating thetemperature gradient distribution inside an overhead wire, andparticularly to a method for determining the amplitude and phase of astratified current of an overhead wire.

BACKGROUND ART

The equation of states of an overhead wire finds, based on thetemperature-tension in a known state, the temperature or tension inanother state, thereby finding the sag of the wire. The equation ofstates simplifies the structural features of an overhead wire during thederivation: it is considered that the entire wire is an isothermal bodyand the cross-sectional stress distribution is uniformly distributed.However, the overhead wire is mostly a steel-cored aluminum strandedwire, which are stranded by several strands of conductors, so that thereis an air gap between the conductors in each layer, of which the heattransfer coefficient is larger relative to the metal conductor, and thetemperature mainly falls in the air, in addition, the heat dissipationcondition of the outer surface is better than that of the inner part, sothe internal temperature of the steel-cored aluminum stranded wires ishigher than the temperature of the outer layer. Thus, the hightemperature range of the wire is borne by the steel core, and the radialtemperature difference can reach more than ten degrees. To this end,accurately calculating the temperature of the steel core or the radialtemperature difference of the steel wire aluminum stranded wire of theoverhead wire will play an important role in improving the calculationaccuracy of such a model.

At present, researchers at home and abroad have done some research onthe radial temperature distribution of the overhead wire, and haveachieved many outstanding outcomes. For example, V. T. Morgan et al.consider the contact thermal resistance of the air gap and the airthermal resistance, and think that the heat generation rate of aconductor is uniformly distributed on a cross section of the conductor.Based on this, the radial temperature calculation formula is deduced indetail. W. Z. Black establishes a heat transfer equation under thecondition that the current is distributed in DC series and parallel. Theradial heat transfer coefficient is divided and valued under differentcurrent loads, different wind speeds and different tension conditions.At home, Ying Zhanfeng et al. establish, in combination with a parameteridentification and thermoelectric analogy method, a radial temperaturethermal path model, which was verified by experiments. However, in theabove review, some simplify the actual structure of the wire, and thinkthat the steel-cored aluminum stranded wire is a coaxial doubleconductor. Although some consider the stranded structure of the wire,the effects of a skin effect on the current distribution and ohmic lossstill have not been considered in calculating the heat generation rateof conductors in each layer, while the above two aspects are mainfactors affecting the existence of the radial gradient. Therefore,accurately calculating the current distribution of the conductors ineach layer within the overhead wire at an AC frequency and the actualheat generation rate of the conductors in each layer will be crucial foraccurately evaluating the temperature of the steel core.

SUMMARY OF THE INVENTION

An object of the present invention is to overcome the above-describeddefects in the prior art. A method for determining the amplitude andphase of a stratified current of an overhead wire is provided.

The object of the present invention can be achieved by taking thefollowing technical solutions:

A method for determining the amplitude and phase of a stratified currentof an overhead wire, the method comprising:

S1, determining the specification, the size and main technicalparameters of a wire, the step specifically being as follows:

S101, determining the number of layers of the overhead wire and thenumber of conductors in each layer and the planned size thereof; and

S102, determining the material of the conductors in each layer and thecorresponding resistivity and magnetic permeability;

S2, calculating mutual inductances between conductors within asingle-phase wire and the self-inductance thereof, the step specificallybeing as follows:

S201, calculating the mutual inductance between a conductor layer i anda conductor layer j within the single-phase wire; and

S202, calculating the self-inductance of the conductor layer i withinthe single-phase wire;

S3, calculating mutual inductance reactance between conductors of athree-phase system and the self-inductance reactance thereof, the stepspecifically being as follows:

S301, calculating the total mutual inductance reactance between theconductor layer i and the conductor layer j within an A-phase wire in athree-phase system; and

S302, calculating the self-inductance reactance of the conductor layer iwithin the A-phase wire in the three-phase system; and

S4, calculating the distribution of currents in each layer.

Further, step S101 is specifically as follows:

numbering the wires and determining the radius of the overhead wire andthe radius of each conductor, wherein each phase of the three-phase wirehas m layers, which are numbered, from inside to outside, as 1, 2, . . .m, there are n conductors in each layer within a wire, no distinction ismade between the conductors in each layer, and the three-phase wires areonly distinguished by subscripts a, b and c in derivation; and

in terms of current, using İ_(i) to indicate the total current of thelayer i, and using İ_(i)′ to indicate the current on a wire in the layeri, that is İ_(i)=nİ_(i)′,

where n is the number of conductors in the layer i, and İ_(i)′ appearsonly in the result analysis to compare effects of a skin effect.

Further, step S102 is specifically as follows:

determining the resistivity and magnetic permeability of variousconductors based on that the overhead wire is a steel-cored aluminumstranded wire, an aluminum stranded wire and a copper wire.

Further, the calculation formula for the mutual inductance M_(aiaj) instep S201 is specifically as follows:

${M_{aiaj} = {\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{ij}} - 1} \right)} \right\rbrack}},{{{wherein}\mspace{14mu} D_{ij}} = \sqrt[{mn}]{\prod\limits_{k = 1}^{m}\; {\prod\limits_{i = 1}^{n}\; \left\lbrack {r_{i}^{2} + r_{j}^{2} - {2r_{i}r_{j}{\cos \left( {\theta_{ik} - \theta_{j\; 1}} \right)}}} \right\rbrack}}},$

where m is the number of conductors in the layer i, n is the number ofconductors in the layer j, D_(ji) is a geometric mean of distancesbetween conductors between the layer i and the layer j, r_(i) is thedistance from the center of circle of a single conductor in the layer ito the center of the wire, r_(j) is the distance from the center ofcircle of the single conductor in the layer j to the center of the wire,and θ_(ik)−θ_(j1) is an opening angle between the center of circle ofthe k^(th) conductor in the layer i and the center of circle of the1^(st) conductor in the layer j, relative to the total center of circleof the wire.

Further, the calculation formula for the self-inductance L_(aiai) in thestep S202 is specifically as follows:

${L_{aiaj} = {\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{ij}} - 1} \right)} \right\rbrack}},{wherein},{D_{ij} = \sqrt[m]{r_{eq}{\prod\limits_{k = 2}^{m}\; \left\lbrack {r_{i}^{2} + r_{1}^{2} - {2r_{i}r_{1}{\cos \left( {\theta_{ik} - \theta_{i1}} \right)}}} \right\rbrack}}}$

where m is the number of conductors in the layer i, D_(ii) is ageometric mean of distances between conductors in the layer i, r_(i) isthe distance from the center of circle of a single conductor in thelayer i to the center of the wire, θ_(ik)−θ_(i1) is an opening anglebetween the center of circle of the k^(th) conductor in the layer i andthe center of circle of the 1^(st) conductor in the layer i, relative tothe total center of circle of the wire, and r_(eq) is an equivalentradius of the first conductor in the layer i.

Further, the step S301 is specifically as follows:

assuming that the system is in three-phase current symmetry, that is

i _(ai) +i _(bi) +i _(ci)=0

the wire is in three-phase symmetry after alternation and the equivalentdistance between wires is D_(eq), and the distance between the wires isconsidered to be much greater than the distance between each strandwithin a one-phase wire, then for the conductor layer i within an Aphase wire, a magnetic flux linkage generated by the current in theconductor layer j is:

$\begin{matrix}{\psi_{aij} = {{M_{aiaj}i_{aj}} + {M_{aibj}i_{bj}} + {M_{aicj}i_{cj}}}} \\{= {{{\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{ij}} - 1} \right)} \right\rbrack}i_{aj}} + {{\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{eq}} - 1} \right)} \right\rbrack}i_{bj}} + {{\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{eq}} - 1} \right)} \right\rbrack}i_{cj}}}} \\{= {{{\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{ij}} - 1} \right)} \right\rbrack}i_{aj}} + {{\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{eq}} - 1} \right)} \right\rbrack}\left( {i_{bj} + i_{cj}} \right)}}} \\{= {\frac{\mu_{0}}{2\pi}{{Ln}\left( \frac{D_{eq}}{D_{ij}} \right)}i_{aj}}}\end{matrix}$

therefore, in a three-phase symmetric system, the total mutualinductance between the conductor layer i within an A-phase wire and theconductor layer j within the A-phase wire is:

$M_{aiaj} = {\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{ij}} - 1} \right)} \right\rbrack}$

and in the three-phase symmetric system, the total mutual inductancereactance between the conductor layer i within the A-phase wire and theconductor layer j within the A-phase wire is:

$X_{aij} = {\mu_{0}f\; {{{Ln}\left( \frac{D_{eq}}{D_{ij}} \right)}.}}$

Further, the step S302 is specifically as follows:

in the mutual inductance reactance

$X_{aij} = {\mu_{0}f\; {{Ln}\left( \frac{D_{eq}}{D_{ij}} \right)}}$

making i=j to obtain the self-inductance reactance of the conductorlayer i

$X_{aii} = {\mu_{0}f\; {{{Ln}\left( \frac{D_{eq}}{D_{ii}} \right)}.}}$

Further, step S4 is specifically as follows:

assuming that in one phase the resistances of each layer from inside tooutside is r₁, r₂, r₃ . . . r_(m) respectively, and taking a wiresegment of unit length, wherein the voltage drops between each layer onthe wire segment should be equal, denoted as V, then there is

V = r₁i₁ + j(X₁₁i₁ + X₁₂i₂ + X₁₃i₃ + …  X_(1 m)i_(m))V = r₂i₂ + j(X₂₁i₁ + X₂₂i₂ + X₂₃i₃ + …  X_(2 m)i_(m))V = r₃i₁ + j(X₃₁i₁ + X₃₂i₂ + X₃₃i₃ + …  X_(3 m)i_(m)) …V = r_(m)i₁ + j(X_(m 1)i₁ + X_(m 2)i₂ + X_(m 3)i₃ + …  X_(m m)i_(m)),

combining the above formulas and eliminating V and D_(eq) to obtain

${{\begin{bmatrix}{r_{1} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{12}}}} \\{r_{1} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{13}}}} \\\ldots \\{r_{1} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{1m}}}}\end{bmatrix}\begin{bmatrix}i_{1} & 0 & 0 & \ldots & 0 \\0 & i_{1} & 0 & \ldots & 0 \\0 & 0 & i_{1} & \ldots & 0 \\\ldots & \; & \; & \; & \; \\0 & 0 & 0 & \ldots & i_{1}\end{bmatrix}} = {\begin{bmatrix}{r_{2} + {j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{22}}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{23}}} & \ldots & {j\; \mu_{0}f\; {Ln}\frac{D_{1m}}{D_{2m}}} \\{j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{23}}} & {r_{3} + {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{23}}}} & \ldots & {j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{3m}}} \\\ldots & \; & \; & \; \\{j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{2\; m}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{3\; m}}} & \ldots & {r_{4} + {j\; \mu_{0}f\; {Ln}\frac{D_{1m}}{D_{mm}}}}\end{bmatrix}\begin{bmatrix}i_{2} \\i_{3} \\\ldots \\i_{m}\end{bmatrix}}},\mspace{20mu} {{{denoting}\mspace{14mu} T} = \begin{bmatrix}{r_{1} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{12}}}} & {r_{1} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{13}}}} & \ldots & {r_{1} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{1m}}}}\end{bmatrix}^{T}}$ $\mspace{20mu} {X = \begin{bmatrix}{r_{2} + {j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{22}}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{23}}} & \ldots & {j\; \mu_{0}f\; {Ln}\frac{D_{1m}}{D_{2m}}} \\{j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{23}}} & {r_{3} + {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{23}}}} & \ldots & {j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{3m}}} \\\ldots & \; & \; & \; \\{j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{2\; m}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{3\; m}}} & \ldots & {r_{4} + {j\; \mu_{0}f\; {Ln}\frac{D_{1m}}{D_{mm}}}}\end{bmatrix}}$ $\mspace{20mu} {{{{then}\mspace{14mu}\begin{bmatrix}i_{2} \\i_{3} \\\ldots \\i_{4}\end{bmatrix}} = {X^{- 1}{T\begin{bmatrix}i_{1} & 0 & 0 & \ldots & 0 \\0 & i_{1} & 0 & \ldots & 0 \\0 & 0 & i_{1} & \ldots & 0 \\\ldots & \; & \; & \; & \; \\0 & 0 & 0 & \ldots & i_{1}\end{bmatrix}}}},}$

when vectors are used for representation

${\begin{bmatrix}{\overset{.}{I}}_{2} \\{\overset{.}{I}}_{3} \\\ldots \\{\overset{.}{I}}_{4}\end{bmatrix} = {X^{- 1}{T\begin{bmatrix}{\overset{.}{I}}_{1} & 0 & 0 & \ldots & 0 \\0 & {\overset{.}{I}}_{1} & 0 & \ldots & 0 \\0 & 0 & {\overset{.}{I}}_{1} & \ldots & 0 \\\ldots & \; & \; & \; & \; \\0 & 0 & 0 & \ldots & {\overset{.}{I}}_{1}\end{bmatrix}}}},$

by means of the solution described above, the ratio distribution betweencurrents of each layer is obtained, and by adding the formula

İ ₁ +İ ₂ +İ ₃ + . . . İ _(m) =İ _(Σ)

the current distribution in each layer is calculated.

Compared with the prior art the present invention has the followingadvantages and effects:

The present invention discloses a method for determining the amplitudeand phase of a stratified current of an overhead wire. In combinationwith the actual structure, specification, size and physical technicalparameters of the LGJ300/40 wire, taking into account theelectromagnetic coupling effects between conductors, the current flowingthrough conductors in each layer is deduced, and the accuracy of themodel is compared by means of an electromagnetic simulation softwareANSOFT MAXWELL.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flowchart of a method for determining the amplitude andphase of a stratified current of an overhead wire disclosed in thepresent invention.

DETAILED DESCRIPTION OF EMBODIMENTS

In order to make the objectives, technical solutions and advantages ofembodiments of the present invention clearer, the technical solutions inembodiments of the present invention will be clearly and completelydescribed below with reference to the accompanying drawings in theembodiments of the present invention. Apparently, the describedembodiments are a part, but not all of the embodiments of the presentinvention. All other embodiments obtained by those of ordinary skill inthe art based on the embodiments of the present invention without anycreative effort shall fall within the protection scope of the presentinvention. On the basis of the embodiments of the present invention, allthe other embodiments obtained by a person skilled in the art withoutany inventive effort shall fall within the scope of protection of thepresent invention.

Embodiment

In this embodiment, in combination with an LGJ300/40 type A phase wirebeing used as an object to be calculated, a method for calculating astratified current of an overhead wire is proposed, however, the methodis not limited to the LGJ300/40 type wire, wherein the 2D cross-sectionview of the LGJ 300/40 type wire consists of four layers, which are,from the inside to the outside, a steel core having a radius of 1.33 mmof which a center of circle is the center of the wire, six steel coreshaving a radius of 1.33 mm of which the centers of circle are uniformlyspaced and distributed in a circle with a radius of 2.66 mm, ninealuminum cores having a radius of 1.995 mm of which the centers ofcircle are uniformly spaced and distributed in a circle with a radius of5.985 mm, and fifteen aluminum cores having a radius of 1.995 mm ofwhich the centers of circle are uniformly spaced and distributed in acircle with a radius of 9.975 mm, respectively.

As in FIG. 1, a flowchart of a method for determining the amplitude andphase of a stratified current of an overhead wire is disclosed, themethod specifically comprising the following steps:

S1, determining the specification, the size and main technicalparameters of a wire, the step specifically further comprising thefollowing sub-steps:

S101, determining the number of layers of the overhead wire and thenumber of conductors in each layer and the planned size thereof;

In a specific embodiment, the 2D cross-section view of the LGJ 300/40type wire consists of four layers, which are, from the inside to theoutside, a steel core having a radius of 1.33 mm of which a center ofcircle is the center of the wire, six steel cores having a radius of1.33 mm of which the centers of circle are uniformly spaced anddistributed in a circle with a radius of 2.66 mm, nine aluminum coreshaving a radius of 1.995 mm of which the centers of circle are uniformlyspaced and distributed in a circle with a radius of 5.985 mm, andfifteen aluminum cores having a radius of 1.995 mm of which the centersof circle are uniformly spaced and distributed in a circle with a radiusof 9.975 mm, respectively.

in terms of current, using İ_(i) to indicate the total current of thelayer i, and using İ_(i)′ to indicate the current on a wire inside thelayer i, that is

İ _(i) =nİ _(i),

where n is the number of conductors in the layer i, and İ_(i)′ appearsonly in the result analysis to compare effects of a skin effect.

S102, determining the material of the conductors in each layer and thecorresponding resistivity and magnetic permeability;

In a specific embodiment, the material of the conductors in the firstand second layers within an overhead wire is steel with a resistivity of5×10-7 Ωm. Since the metal steel is a ferromagnetic material which willchange as the current changes, and the value of relative magneticpermeability varies from 1 to 2000. The material of the conductors inthe third and fourth layers is aluminum with a resistivity of 2.83×10-8Ωm, which is a non-ferromagnetic material, and the value of the relativemagnetic permeability is 1.0.

S2, calculating mutual inductances between conductors within an A-phasewire and the self-inductance thereof, the step specifically comprisingthe following sub-steps:

S201, calculating the mutual inductance between a conductor layer i anda conductor layer j within an A-phase wire;

${M_{aiaj} = {\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{ij}} - 1} \right)} \right\rbrack}},{{{wherein}\mspace{14mu} D_{ij}} = \sqrt[{mn}]{\prod\limits_{k = 1}^{m}\; {\prod\limits_{i = 1}^{n}\; \left\lbrack {r_{i}^{2} + r_{j}^{2} - {2r_{i}r_{j}{\cos \left( {\theta_{ik} - \theta_{j\; 1}} \right)}}} \right\rbrack}}},$

where m is the number of conductors in the layer i, n is the number ofconductors in the layer j, D_(ij) is a geometric mean of distancesbetween conductors between the layer i and the layer j, r_(i) is thedistance from the center of circle of a single conductor in the layer ito the center of the wire, r_(j) is the distance from the center ofcircle of the single conductor in the layer j to the center of the wire,and θ_(jk)−θ_(j1) is an opening angle between the center of circle ofthe k^(th) conductor in the layer i and the center of circle of the1^(st) conductor in the layer j, relative to the total center of circleof the wire.

S202, calculating the self-inductance of the conductor layer i withinthe A-phase wire;

${L_{aiaj} = {\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{ij}} - 1} \right)} \right\rbrack}},{wherein},{D_{ii} = \sqrt[m]{r_{eq}{\prod\limits_{k = 2}^{m}\; \left\lbrack {r_{i}^{2} + r_{1}^{2} - {2r_{i}r_{1}{\cos \left( {\theta_{ik} - \theta_{i\; 1}} \right)}}} \right\rbrack}}},$

where m is the number of conductors in the layer i, D_(ii) is ageometric mean of distances between conductors in the layer i, r_(i) isthe distance from the center of circle of a single conductor in thelayer i to the center of the wire, θ_(ik)−θ_(i1) is an opening anglebetween the center of circle of the k^(th) conductor in the layer i andthe center of circle of the 1^(st) conductor in the layer i, relative tothe total center of circle of the wire, and r_(eq) is an equivalentradius of the first conductor in the layer i.

S3, calculating mutual inductance reactance between conductors withineach single-phase wire of a three-phase system and the self-inductancereactance thereof, the step specifically comprising the followingsub-steps:

S301, calculating the total mutual inductance reactance between theconductor layer i and the conductor layer j within an A-phase wire in athree-phase system.

assuming that the system is in three-phase current symmetry, that is

İ ₁ +İ ₂ +İ ₃ + . . . İ _(m) =İ _(Σ),

the wire is in three-phase symmetry after alternation and the equivalentdistance between wires is D_(eq), and the distance between the wires isconsidered to be much greater than the distance between each strandwithin a one-phase wire, then for the conductor layer i within an Aphase wire, a magnetic flux linkage generated by the current in theconductor layer j is:

$\begin{matrix}{\psi_{aij} = {{M_{aiaj}i_{aj}} + {M_{aibj}i_{bj}} + {M_{aicj}i_{cj}}}} \\{= {{{\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{ij}} - 1} \right)} \right\rbrack}i_{aj}} + {{\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{eq}} - 1} \right)} \right\rbrack}i_{bj}} + {{\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{eq}} - 1} \right)} \right\rbrack}i_{cj}}}} \\{= {{{\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{ij}} - 1} \right)} \right\rbrack}i_{aj}} + {{\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{eq}} - 1} \right)} \right\rbrack}\left( {i_{bj} + i_{cj}} \right)}}} \\{= {\frac{\mu_{0}}{2\pi}{{Ln}\left( \frac{D_{eq}}{D_{ij}} \right)}i_{aj}}}\end{matrix}$

in a three-phase symmetric system, the total mutual inductance betweenthe conductor layer i within an A-phase wire and the conductor layer jwithin the A-phase wire is

$M_{aiaj} = {\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{ij}} - 1} \right)} \right\rbrack}$

and in the three-phase symmetric system, the total mutual inductancereactance between the conductor layer i within the A-phase wire and theconductor layer j within the A-phase wire is

${X_{aij} = {\mu_{0}f\; {{Ln}\left( \frac{D_{eq}}{D_{ij}} \right)}}},$

where f is the grid frequency, and since the system is in three-phasesymmetry, X_(ij) is used afterwards to represent the mutual inductancereactance between a layer i and a layer j of a certain phase, that is

$X_{aij} = {\mu_{0}f\; {{{Ln}\left( \frac{D_{eq}}{D_{ij}} \right)}.}}$

S302, calculating the self-inductance reactance of the conductor layer iwithin the A-phase wire in the three-phase system; and

In the above formula, by making i=j, the self-inductance reactance ofthe conductor layer i can be obtained

$X_{aii} = {\mu_{0}f\; {{{Ln}\left( \frac{D_{eq}}{D_{ii}} \right)}.}}$

S4, calculating the distribution of currents in each layer.

assuming that in one phase the resistances of each layer from inside tooutside is r₁, r₂, r₃ . . . r₄ respectively, and taking a wire segmentof unit length, wherein the voltage drops between each layer on the wiresegment should be equal, denoted as V, then there is

V=r ₁ i ₁ +j(X ₁₁ i ₁ +X ₁₂ i ₂ +X ₁₃ i ₃ +X ₁₄ i ₄)

V=r ₂ i ₂ +j(X ₂₁ i ₁ +X ₂₂ i ₂ +X ₂₃ i ₃ +X ₂₄ i ₄)

V=r ₃ i ₃ +j(X ₃₁ i ₁ +X ₃₂ i ₂ +X ₃₃ i ₃ +X ₃₄ i ₄)

V=r ₄ i ₄ +j(X ₄₁ i ₁ +X ₄₂ i ₂ +X ₄₃ i ₃ +X ₄₄ i ₄),

combining the above formulas and eliminating V and D_(eq) to obtain

${\begin{bmatrix}{r_{1} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{12}}}} \\{r_{1} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{13}}}} \\{{.r_{1}} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{14}}}}\end{bmatrix}\begin{bmatrix}i_{1} & 0 & 0 \\0 & i_{1} & 0 \\0 & 0 & i_{1}\end{bmatrix}} = {\quad{{\begin{bmatrix}{r_{2} + {j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{22}}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{23}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{14}}{D_{24}}} \\{j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{23}}} & {r_{3} + {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{23}}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{34}}} \\{j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{24}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{34}}} & {r_{4} + {j\; \mu_{0}f\; {Ln}\frac{D_{14}}{D_{44}}}}\end{bmatrix}\begin{bmatrix}i_{2} \\i_{3} \\i_{4}\end{bmatrix}},\mspace{20mu} {{{denoting}\mspace{20mu} T} = \left\lbrack {r_{1} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{12}}\mspace{14mu} r_{1}} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{13}}\mspace{14mu} r_{1}} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{14}}}} \right\rbrack^{T}},\mspace{20mu} {X = \begin{bmatrix}{r_{2} + {j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{22}}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{23}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{14}}{D_{24}}} \\{j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{23}}} & {r_{3} + {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{23}}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{34}}} \\{j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{24}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{34}}} & {r_{4} + {j\; \mu_{0}f\; {Ln}\frac{D_{14}}{D_{44}}}}\end{bmatrix}},\mspace{20mu} {{{then}\mspace{20mu}\begin{bmatrix}i_{2} \\i_{3} \\i_{4}\end{bmatrix}} = {X^{- 1}{T\begin{bmatrix}i_{1} & 0 & 0 \\0 & i_{1} & 0 \\0 & 0 & i_{1}\end{bmatrix}}}},}}$

when vectors are used for representation

${\begin{bmatrix}{\overset{.}{I}}_{2} \\{\overset{.}{I}}_{3} \\{\overset{.}{I}}_{4}\end{bmatrix} = {X^{- 1}{T\begin{bmatrix}{\overset{.}{I}}_{1} & 0 & 0 \\0 & {\overset{.}{I}}_{1} & 0 \\0 & 0 & {\overset{.}{I}}_{1}\end{bmatrix}}}},$

by means of the solution described above, the ratio between currents ofvarious layers is obtained, and by adding the formula

İ ₁ +İ ₂ +İ ₃ +İ ₄ =İ _(Σ)

the current distribution in each layer is calculated.

Model Effect Analysis:

Using the model calculation process described above, the current of eachlayer within the LGJ300/40 type wire is calculated, and the totaleffective value of the applied current is 700 A, and the phase angle is0°. Comparing the calculated results with the finite element calculationresults, the comparison results are shown in Table 1:

TABLE 1 Comparison table of calculation results the first layer thesecond layer the third layer the fourth layer finite İ₁′ = 0.65 ∠ −29.0İ₂′ = 0.67 ∠ −20.9 İ₃′ = 28.91 ∠ −5.0 İ₄ ′ = 29.17 ∠ +4.0 element μ_(r)= 1000 İ₁′ = 0.74 ∠ −16.3 İ₂′ = 0.74 ∠ −16.2 İ₃′ = 28.93 ∠ −4.5 İ₄′ =29.12 ∠ +2.9 error % 13.31 10.51 0.08 −0.19 μ_(r) = 2000 İ₁′ = 0.70 ∠−25.8 İ₂′ = 0.70 ∠ −25.7 İ₃′ = 28.95 ∠ −4.5 İ₄′ = 29.14 ∠ +2.9 error % 8.32  5.67 0.12 −0.11

Considering that the steel-cored aluminum stranded wire has anon-uniform current distribution at the power frequency, and the currentmainly flows through the aluminum conductor layer, so that the heatgeneration inside the conductors mainly occurs in the aluminum conductorlayer. Although the calculation result of the patent of the presentinvention differs greatly in the simulation results of the steel-coredconductor and the finite element, for the aluminum conductor layerhaving a large heat generation rate, by correcting the relative magneticpermeability, the error can be reduced to 0.125%. In addition, themethod can reflect the phase difference between the conductors in eachlayer. Therefore, when calculating the internal radial temperaturedistribution of an overhead wire or calculating the temperature of asteel core, the calculation method in this patent can be used tocalculate the current distribution and the heat generation rate of eachlayer.

The above-described embodiments are preferred embodiments of the presentinvention; however, the embodiments of the present invention are notlimited to the above-described embodiments, and any other change,modification, replacement, combination, and simplification made withoutdeparting from the spirit, essence, and principle of the presentinvention should be an equivalent replacement and should be includedwithin the scope of protection of the present invention.

1. A method for determining the amplitude and phase of a stratifiedcurrent of an overhead wire, characterized in that the method comprises:S1, determining the specification, the size and main technicalparameters of a wire, the step specifically being as follows: S101,determining the number of layers of the overhead wire and the number ofconductors in each layer and the planned size thereof; and S102,determining the material of the conductors in each layer and thecorresponding resistivity and magnetic permeability; S2, calculatingmutual inductances between conductors within a single-phase wire and theself-inductance thereof, the step specifically being as follows: S201,calculating the mutual inductance between a conductor layer i and aconductor layer j within the single-phase wire; and S202, calculatingthe self-inductance of the conductor layer i within the single-phasewire; S3, calculating mutual inductance reactance between conductorswithin the single-phase wire in a three-phase system and theself-inductance reactance thereof, the step specifically being asfollows: S301, calculating the total mutual inductance reactance betweenthe conductor layer i and the conductor layer j within an A-phase wirein a three-phase system; and S302, calculating the self-inductancereactance of the conductor layer i within the A-phase wire in thethree-phase system; and S4, calculating the distribution of currents inconductors in each layer within the single-phase wire.
 2. The method fordetermining the amplitude and phase of a stratified current of anoverhead wire of claim 1, characterized in that step S101 isspecifically as follows: numbering the wires and determining the radiusof the overhead wire and the radius of each conductor, wherein eachphase of the three-phase wire has m layers, which are numbered, frominside to outside, as 1, 2, . . . m, there are n conductors in eachlayer within a wire, no distinction is made between conductors in eachlayer, and the three-phase wires are only distinguished by subscripts a,b and c in derivation; and in terms of current, using İ_(i) to indicatethe total current of the layer i, and using İ_(i)′ to indicate thecurrent on a conductor in the layer i, that is İ_(i)=nİ_(i)′, where n isthe number of conductors in the layer i, and İ_(i)′ appears only in theresult analysis to compare effects of a skin effect.
 3. The method fordetermining the amplitude and phase of a stratified current of anoverhead wire of claim 1, characterized in that step S102 isspecifically as follows: determining the resistivity and magneticpermeability of various conductors based on if the overhead wire is asteel-cored aluminum stranded wire, an aluminum stranded wire or acopper wire.
 4. The method for determining the amplitude and phase of astratified current of an overhead wire of claim 1, characterized in thatthe calculation formula for the mutual inductance M_(aiaj) in step S201is specifically as follows:${M_{aiaj} = {\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{ij}} - 1} \right)} \right\rbrack}},{{{wherein}\mspace{14mu} D_{ij}} = \sqrt[{mn}]{\prod\limits_{k = 1}^{m}{\prod\limits_{i = 1}^{n}\left\lbrack {r_{i}^{2} + r_{j}^{2} - {2r_{i}r_{j}{\cos \left( {\theta_{ik} - \theta_{j\; 1}} \right)}}} \right\rbrack}}},$where m is the number of conductors in the layer i, n is the number ofconductors in the layer j, D_(ij) is a geometric mean of distancesbetween conductors located in the layer i and the layer j respectively,r_(i) is the distance from the center of circle of a single conductor inthe layer i to the center of the wire, r_(j) is the distance from thecenter of circle of the single conductor in the layer j to the center ofthe wire, and θ_(ik)−θ_(j1) is an opening angle between the center ofcircle of the k^(th) conductor in the layer i and the center of circleof the 1^(st) conductor in the layer j, relative to the center of circleof the wire.
 5. The method for determining the amplitude and phase of astratified current of an overhead wire of claim 1, characterized in thatthe calculation formula for the self-inductance L_(aiai) in step S202 isspecifically as follows:${L_{aiaj} = {\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{ij}} - 1} \right)} \right\rbrack}},{wherein},{D_{ii} = \sqrt[m]{r_{eq}{\prod\limits_{k = 2}^{m}\left\lbrack {r_{i}^{2} + r_{1}^{2} - {2r_{i}r_{1}{\cos \left( {\theta_{ik} - \theta_{i\; 1}} \right)}}} \right\rbrack}}}$where m is the number of conductors in the layer i, D_(ii) is ageometric mean of distances between conductors in the layer i, r_(i) isthe distance from the center of circle of a single conductor in thelayer i to the center of the wire, θ_(ik)−θ_(i1) is an opening anglebetween the center of circle of the k^(th) conductor in the layer i andthe center of circle of the 1^(st) conductor in the layer i, relative tothe total center of circle of the wire, and r_(eq) is an equivalentradius of the first conductor in the layer i.
 6. The method fordetermining the amplitude and phase of a stratified current of anoverhead wire of claim 1, characterized in that step S301 isspecifically as follows: assuming that the system is in three-phasecurrent symmetry, that isi _(ai) +i _(bi) +i _(ci)=0 the wire is in three-phase symmetry afteralternation and the equivalent distance between wires is D_(eq), and thedistance between the wires is much greater than the distance betweeneach strand within a one-phase wire, then for the conductor layer iwithin an A phase wire, a magnetic flux linkage generated by the currentin the conductor layer j is: $\begin{matrix}{\Psi_{aij} = {{M_{aiaj}i_{aj}} + {M_{aibj}i_{bj}} + {M_{aicj}i_{cj}}}} \\{= {{{\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{ij}} - 1} \right)} \right\rbrack}i_{aj}} + {{\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{eq}} - 1} \right)} \right\rbrack}i_{bj}} + {{\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{eq}} - 1} \right)} \right\rbrack}i_{cj}}}} \\{= {{{\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{ij}} - 1} \right)} \right\rbrack}i_{aj}} + {{\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{eq}} - 1} \right)} \right\rbrack}\left( {i_{bj} + i_{cj}} \right)}}} \\{= {\frac{\mu_{0}}{2\pi}{{Ln}\left( \frac{D_{eq}}{D_{ij}} \right)}i_{aj}}}\end{matrix}$ then, in a three-phase symmetric system, the total mutualinductance between the conductor layer i within an A-phase wire and theconductor layer j within the A-phase wire is:$M_{aiaj} = {\frac{\mu_{0}}{2\pi}\left\lbrack {{Ln}\left( {\frac{2l}{D_{ij}} - 1} \right)} \right\rbrack}$and in the three-phase symmetric system, the total mutual inductancereactance between the conductor layer i within the A-phase wire and theconductor layer j within the A-phase wire is:$X_{aij} = {\mu_{0}f\; {{{Ln}\left( \frac{D_{eq}}{D_{ij}} \right)}.}}$7. The method for determining the amplitude and phase of a stratifiedcurrent of an overhead wire of claim 6, characterized in that step S302is specifically as follows: in the mutual inductance reactance$X_{aij} = {\mu_{0}f\; {{Ln}\left( \frac{D_{eq}}{D_{ij}} \right)}}$making i=j to obtain the self-inductance reactance of the conductorlayer i$X_{aii} = {\mu_{0}f\; {{{Ln}\left( \frac{D_{eq}}{D_{ii}} \right)}.}}$8. The method for determining the amplitude and phase of a stratifiedcurrent of an overhead wire of claim 1, characterized in that step S4 isspecifically as follows: assuming that in one phase the resistances ofeach layer from inside to outside is r₁, r₂, r₃ . . . r_(m)respectively, and taking a wire segment of unit length, wherein thevoltage drops between each layer on the wire segment should be equal,denoted as V, then there isV = r₁i₁ + j(X₁₁i₁ + X₁₂i₂ + X₁₃i₃ + …  X_(1m)i_(m))V = r₂i₂ + j(X₂₁i₁ + X₂₂i₂ + X₂₃i₃ + …  X_(2m)i_(m))V = r₃i₁ + j(X₃₁i₁ + X₃₂i₂ + X₃₃i₃ + …  X_(3m)i_(m)) …V = r_(m)i₁ + j(X_(m 1)i₁ + X_(m 2)i₂ + X_(m 3)i₃ + …  X_( m m)i_(m)),combining the above formulas and eliminating V and D_(eq) to obtain${\begin{bmatrix}{r_{1} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{12}}}} \\{r_{1} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{13}}}} \\\ldots \\{r_{1} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{1m}}}}\end{bmatrix}\begin{bmatrix}i_{1} & 0 & 0 & \ldots & 0 \\0 & i_{1} & 0 & \ldots & 0 \\0 & 0 & i_{1} & \ldots & 0 \\\ldots & \; & \; & \; & \; \\0 & 0 & 0 & \ldots & i_{1}\end{bmatrix}} = {\quad{{\begin{bmatrix}{r_{2} + {j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{22}}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{23}}} & \ldots & {j\; \mu_{0}f\; {Ln}\frac{D_{1m}}{D_{2m}}} \\{j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{23}}} & {r_{3} + {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{23}}}} & \ldots & {j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{3m}}} \\\ldots & \; & \; & \; \\{j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{2m}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{3m}}} & \ldots & {r_{4} + {j\; \mu_{0}f\; {Ln}\frac{D_{1m}}{D_{m\; m}}}}\end{bmatrix}\begin{bmatrix}i_{2} \\i_{3} \\\ldots \\i_{m}\end{bmatrix}},\mspace{20mu} {T = {{\left\lbrack {r_{1} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{12}}\mspace{14mu} r_{1}} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{13}}\mspace{14mu} \ldots \mspace{14mu} r_{1}} - {j\; \mu_{0}f\; {Ln}\frac{D_{11}}{D_{1m}}}} \right\rbrack^{T}\mspace{20mu} {denoting}X} = {{\left\lbrack \begin{matrix}{r_{2} + {j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{22}}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{23}}} & \ldots & {j\; \mu_{0}f\; {Ln}\frac{D_{1m}}{D_{2m}}} \\{j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{23}}} & {r_{3} + {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{23}}}} & \ldots & {j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{3m}}} \\\ldots & \; & \; & \; \\{j\; \mu_{0}f\; {Ln}\frac{D_{12}}{D_{2m}}} & {j\; \mu_{0}f\; {Ln}\frac{D_{13}}{D_{3m}}} & \ldots & {r_{4} + {j\; \mu_{0}f\; {Ln}\frac{D_{1m}}{D_{m\; m}}}}\end{matrix} \right\rbrack \mspace{20mu} {{then}\mspace{14mu}\begin{bmatrix}i_{2} \\i_{3} \\\ldots \\i_{4}\end{bmatrix}}} = {X^{- 1}{T\begin{bmatrix}i_{1} & 0 & 0 & \ldots & 0 \\0 & i_{1} & 0 & \ldots & 0 \\0 & 0 & i_{1} & \ldots & 0 \\\ldots & \; & \; & \; & \; \\0 & 0 & 0 & \ldots & i_{1}\end{bmatrix}}}}}},}}$ when vectors are used for representation${\begin{bmatrix}{\overset{.}{I}}_{2} \\{\overset{.}{I}}_{3} \\\ldots \\{\overset{.}{I}}_{4}\end{bmatrix} = {X^{- 1}{T\begin{bmatrix}{\overset{.}{I}}_{1} & 0 & 0 & \ldots & 0 \\0 & {\overset{.}{I}}_{1} & 0 & \ldots & 0 \\0 & 0 & {\overset{.}{I}}_{1} & \ldots & 0 \\\ldots & \; & \; & \; & \; \\0 & 0 & 0 & \ldots & {\overset{.}{I}}_{1}\end{bmatrix}}}},$ by means of the solution described above, the ratiodistribution between currents of each layer is obtained, and by addingthe formulaİ ₁ +İ ₂ +İ ₃ + . . . İ _(m) =İ _(Σ) the current distribution in eachlayer is calculated.